A necessary trait of all living organisms is their adaptability. They are governed by processes that adapt to external/non-genetic perturbations (homeostasis) and are robust to internal mutational/genetic changes (referred to here as robustness, for simplicity). Mathematically, a priori, these two concepts are unrelated: Natural selection is determined by the survival and reproductive capabilities, in particular homeostasis, of an individuals phenotype. Homeostasis is a dynamic property. On the other hand, germline mutations change the genotype of progeny without affecting the fitness of the parent. Robustness is, however, necessary for an individuals phenotype to be transmitted to progeny, and is a static property. In fact, many biological networks that protect the organism against changes in the environment are also able to prevent phenotypic changes due to genetic mutation. The aim of this study is to investigate the effect of network topologies on their capacity for adaptation and to arrive at a quantitative description of the relationship between homeostasis and robustness. We analyzed all possible topologies of three-node enzymatic networks using a quantitative measure of the degree of robustness to both input and parameter perturbations. After selecting for adaptable networks using the Pearson correlation, we found a strong statistical correlation between adaptable homeostasis and robustness. We investigated the topological motifs that are necessary and/or sufficient to achieve adaptability, homeostasis, and/or robustness. Many researchers in quantitative biology face the challenge of analyzing large datasets in order to identify the molecular interactions underlying an observed function in a biological system. This identification is made difficult due to the robustness of many biological functions to both environmental and genetic mutations. Though a relationship between these two features has been extensively investigated in the context of evolution or population genetics, network structures or architectures that imply a correlation have never been investigated in a general theoretical context. In this study, we show that there is a statistical correlation between the two types of robustness by testing a large number of dynamic networks of ordinary differential equations ranging up to 30 interacting nodes. What is particularly interesting is that this correlation is stronger in the subset of networks that show a transient response to perturbations, and becomes stronger as the size of the networks increases. We show how to use this observation to find networks that display both features and refine the classification of known 3-node motifs in terms of their environmental and genetic robustness values. This work is an essential stepping stone towards a quantitative and predictive understanding of the interplay between homeostasis and robustness in natural selection.